 # Arithmetic or Geometric? a) If a_{1}, a_{2}, a_{3}, \cdots is an usagirl007A 2021-08-13 Answered
Arithmetic or Geometric?
a) If ${a}_{1},{a}_{2},{a}_{3},\cdots$ is an arithmetic sequence, is the sequence ${a}_{1}+2,{a}_{2}+2,{a}_{3}+2,\cdots$ arithmetic?
b) If ${a}_{1},{a}_{2},{a}_{3},\cdots$ is a geometric sequence, is the sequence $5{a}_{1},5{a}_{2},5{a}_{3},\cdots$ geometric?
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a) To show: the sequence ${a}_{1}+2,{a}_{2}+2,{a}_{3}+2,\cdots$ is arithmetic.
Given:
The sequence ${a}_{1},{a}_{2},{a}_{3},\cdots$ is an arithmetic sequence.
Approach:
In arithmetic sequence, difference between the terms remains constant throughout.
Calculation:
The sequence ${a}_{1},{a}_{2},{a}_{3},\cdots$ is an arithmetic sequence.
${a}_{2}-{a}_{1}={a}_{3}-{a}_{2}=\cdots \cdots \left(1\right)$
$\left({a}_{2}+2\right)-\left({a}_{1}+2\right)=\left({a}_{2}-{a}_{1}\right)$
$\left({a}_{3}+2\right)-\left({a}_{2}+2\right)=\left({a}_{3}-{a}_{2}\right)$
From equation (1)
$\left({a}_{3}+2\right)-\left({a}_{2}+2\right)=\left({a}_{2}-{a}_{1}\right)$
As the differences between the terms are same then it is an arithmetic sequence.
Conclusion: hence, the sequence ${a}_{1}+2,{a}_{2}+2,{a}_{3}+2,\cdots$ is arithmetic.
b) To show: $5{a}_{1},5{a}_{2},5{a}_{3},\cdots$ is a geometric sequence.
Given: The sequence ${a}_{1},{a}_{2},{a}_{3},\cdots$ is geometric sequence.
Approach: In a geometric sequence each term is found by multiplying the previous term by a constant.
Calculation:
$\frac{{a}_{2}}{{a}_{1}}=\frac{{a}_{3}}{{a}_{2}}=\cdots \cdots \left(1\right)$
Multiplying equation (1) by 5
$\frac{5{a}_{2}}{5{a}_{1}}=\frac{5{a}_{3}}{5{a}_{2}}=\cdots$
Then it is a geometric sequence.
Conclusion: hence, the sequence $5{a}_{1},5{a}_{2},5{a}_{3},\cdots$ is a geometric sequence.